Datasheet
AN3180 Electrical equivalent circuit models of coupled inductors and transformers
Doc ID 17273 Rev 1 31/39
Figure 24. Electrical equivalent circuit of coupled inductors
The branch-constitutive equations of the circuit are the following:
Equation 25
By comparing Equation 25 to 21 it is possible to find the following relationships:
Equation 26
It is important to notice that the model (Equation 25) and the resulting relationships
(
Equation 26) use four parameters (L
µ
, L
a
, L
b
, a), but equations (Equation 21) show that
three parameters only (L
1
, L
2
, M) are needed to completely define the two-port circuit. This
means that one of the four parameters in (
Equation 25) - a is the obvious choice - can be
arbitrarily fixed, therefore leading to an infinite number of models (
Equation 25) equivalent to
(
Equation 21).
A good criterion for choosing a is that both L
a
and L
b
have a positive value: should they
result otherwise, the terminal equations would still be represented correctly but a negative
inductance does not make physical sense and leads to wrong results as far as energy
considerations are concerned.
It is possible to show that, if a equals the secondary-to-primary turns ratio
n=N
2
/N
1
, L
a
and
L
b
are always positive. Moreover, it is possible to prove that this choice leads to the same
model as the reluctance model approach; and so the model with a =
n is the physical model
of a coupled inductor.
L
µ
is associated to the mutual flux that links the primary and secondary winding mostly
through the magnetic core, it is called primary magnetizing inductance and is designated by
L
M
; L
a
is associated to the flux generated by the primary winding and not completely linked
to itself or to the secondary winding, that is the primary leakage flux: L
a
is therefore called
primary leakage inductance and is designated by L
l1
. Similarly, on the secondary side the
!-V
L
W
L
W
0
/
/
Y
W Y
W
L
LW
W
D
Y
W Y
W/
/
D /
E
LGHDO
L
W
Y
W Y
W
L
W L
W
(t)
(t)
dt
d
LLL
LLL
(t)
(t)
b
2
a
2
1
2
1
i
i
v
v
+
+
=
μμ
μμ
aa
a
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
−=
=
−=
⇒
−=
=
−=
⇒
⎪
⎩
⎪
⎨
⎧
+=
=
+=
μ
μ
μ
μ
μ
μ
μ
MLL
M
L
M
LL
LLL
M
L
LLL
LLL
LM
LLL
2b
1a
2
2b
1a
b
2
2
a1
a
a
a
a
a
a
a